Robust control in a rough environment

Abstract

This paper studies robust decision problems in a rough environment that is described by Volterra processes. Various alternative dominated models are introduced to reflect decision makers' concerns regarding model uncertainty. Using a functional Itô calculus approach, we characterize the robust optimal strategy by a path-dependent Hamilton–Jacobi–Bellman–Isaacs equation. Explicit strategies are derived for robust power and exponential utility maximization with the Volterra Heston model and an example from a robust linear-quadratic control problem. Numerical study shows that it becomes harder to distinguish two probability measures in a rougher environment. Robust optimal investment strategies can reduce losses from ignoring volatility roughness and model uncertainty and can improve the stability of portfolio performance.